128 XIII INTEGRATION (5) there exists an integrable function g± g: 0 such that, for all z e E, we have \D2f(x, z)\ 5* 9i(x) almost everywhere in X. Then h is differentiable at every point ofE, and we have (13.8.6.1) h'(z) = f D2/(x, z) dtfx) ("differentiation under the.integral sign"). (iii) Now suppose that E is an open set in C and that f is a mapping of X x E into C, satisfying the following conditions: (6) for almost all x e X, the function zi—>f(x, z) is analytic in E; (7) for all z e E, the function x H-»/(JC, z) is integrable ; (8) there exists an integrable function g^O such that for all z e E we have \f(x9 z)| <j! g(x) almost everywhere in X. Then h is analytic in E, and for each integer k > 0 and each point z e E, the function x\~* D*/(^, z) fa integrable, and we have (i) By (3.13.14) it is enough to show that, for every sequence (zn) tending to z0 in E, we have lim h(zn) = A(z0); and this follows from the hypotheses B-^OO and from (13.8.4). (ii) Again by virtue of (3.13.14), it is enough to show that, for each sequence (O of real numbers =£0 and tending to 0, the sequence has as limit the right-hand side of (13.8.6.1). But it follows from the hypotheses and from (8.5.4) that for all n} so that the result again follows from (13.8.4). (iii) Let y : t\-+a + relt (0 <; t g 2n) be a circuit contained in E. For each z in the interior of the disc |z — a\ <£ r, we have (13.8.6.2) f(x, z) = — I •/V*'M V / «/ V^J **/ /^__- I f ~~" Zen h(z) = \f(x, z) dfi(x) is continuous at ZQ .